[meindnoch]

Homomorphically encrypted services don't need a priori knowledge of the encryption key. That's literally the whole point.

Consider the following (very weak) encryption scheme:

m, k ∈ Z[p], E(m) = m * k mod p, D(c) = c * k⁻¹ mod p

With this, I can implement a service that receives two cyphertexts and computes their encrypted sum, without knowledge of the key k:

E(x) + E(y) = x * k + y * k mod p = (x + y) * k mod p = E(x + y)

Of course, such a service is not too interesting, but if you could devise an algebraic structure that supported sufficiently complex operations on cyphertexts (and with a stronger encryption), then by composing these operations one could implement arbitrarily complex computations.

[yorwba]

In the case of searching Google, E(x) is the encrypted query and y is Google's database. Can you compute E(x + y) without doing at least as much work as computing E(y)? I don't think so. Instead, you use public key cryptography so that the server can compute E(y) (yes, encrypting the entire database) without being able to decrypt D(E(x)) = x.

[meindnoch]

Wrong. In the case of searching, the database is the function that you feed the input query into.

E.g. consider the following system:

E(x) = x ^ k, D(x) = x ^ k

So a one-time pad. Let's say that I provide a service that lets you decide whether a number is even or odd:

IsOdd(E(x)) = E(x) mod 2

You give it an encrypted number, and it gives you back an encrypted bit that you can decrypt to see if the original number was even or not. All while it has zero knowledge of your plaintext number.

A homomorphically encrypted database is just like IsOdd(x), except orders of magnitude more complex. One idea is that any computation can be turned into a Boolean circuit, so if you have homomorphic building blocks for Boolean circuits, you can implement any computation. Obviously, some caveats apply, like all loops have to be unrolled, etc. That's why the whole thing is so inefficient. But mathematically it works.

[yorwba]

If I'm generous, that "database" stores a single number. If you transform a more realistic database (one that can store many arbitrary values) into a Boolean circuit, you'll generally end up with at least one operation per stored value. And when you evaluate the circuit on encrypted data, you have to evaluate all those operations every time. That's why I wrote "without doing at least as much work as computing E(y)" instead of just "without computing E(y)." Yes, you do not necessarily explicitly encrypt all stored values. But you'll end up performing a corresponding amount of computation anyways.

[teo_zero]

> IsOdd(E(x)) = E(x) mod 2

I don't get this. Did you mean

IsOdd(E(x)) = x mod 2

But who provides such function? In the case of a web search,the function is the search engine. I expect Google to provide it, not the user: the refinement and completeness level of its engine is the only reason to choose Google over its competitor. If I have to provide the function, then I'm only buying the compute power from them.

[meindnoch]

>I don't get this. Did you mean

>IsOdd(E(x)) = x mod 2

No, that's literally the point. IsOdd() operates on the cyphertext E(x). It doesn't see the plaintext x. And yet, due to its algebraic properties, the server can answer the query without decrypting it.

For example:

  Client:
    x = 13
    k = 45
    E(x) = 13 xor 45 = 32

  Server:
    IsOdd(E(x)) = 32 mod 2 = 0

  Client:
    D(IsOdd(E(x))) = D(0) = 0 xor TrimLength(45, 1) = 1 (we trim the decryption key to the length of the response, which is 1bit)

So what happens in this case is the server sends you a bit, which you'll either flip or not flip, depending on the key k. The server doesn't know whether you're going to flip its answer or not, so it doesn't know if your plaintext number is odd or even.

[teo_zero]

Now I see it. Thank you.

What I still don't understand is how this can be used to have the remote host give me responses that I couldn't have calculated myself locally, because all the data is stored in the cloud, not locally (that's how Google search works).

[meindnoch]

Well, consider a function that is vastly more complex than IsOdd() from my previous example. IsOdd() was a function from Z[2]^n to Z[2]. Instead, imagine a function from Z[2]^n to Z[2]^m, aka a general N-to-M bit Boolean function. Any such Boolean function can be represented as a logic circuit, that is, AND/OR/NOT gates and connections between them. Now, if we can implement these logic gates (and the connections between them) homomorphically, then we can implement a homomorphic realization of arbitrary functions. One "small" caveat when transforming programs into logic circuits is that all data must be inlined, all loops must be unrolled, all branches must be executed.

[bcrl]

It's an excellent way of upholding Wirth's Law: software will continue to get slower more rapidly than hardware becomes faster.